\[ \int \frac {(d+e x)^{3/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (-b e g +c d g +c e f \right ) \left (e x +d \right )^{\frac {3}{2}}}{c \,e^{2} \left (-b e +2 c d \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}+\frac {2 \left (-2 b e g +3 c d g +c e f \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{c^{2} e^{2} \left (-b e +2 c d \right ) \sqrt {e x +d}} \]
command
integrate((e*x+d)^(3/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} g e^{\left (-2\right )}}{c^{2}} + \frac {2 \, {\left (c d g + c f e - b g e\right )} e^{\left (-2\right )}}{\sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{2}} - \frac {2 \, {\left (3 \, c d g + c f e - 2 \, b g e\right )} e^{\left (-2\right )}}{\sqrt {2 \, c d - b e} c^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________